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📥 Download PDF (Free)Chapter 6 - Applications of Derivatives - covers rate of change, tangents/normals, increasing/decreasing functions, maxima/minima, and approximations. Carries 8-10 marks.
Key Concepts
Rate of Change
dy/dx represents rate of change of y with respect to x.
If s = f(t) is displacement, then ds/dt = velocity, dยฒs/dtยฒ = acceleration
If s = f(t) is displacement, then ds/dt = velocity, dยฒs/dtยฒ = acceleration
Tangent and Normal
Slope of tangent at (xโ,yโ): m = dy/dx at (xโ,yโ)
Equation of tangent: y โ yโ = m(x โ xโ)
Slope of normal = โ1/m
Equation of normal: y โ yโ = (โ1/m)(x โ xโ)
Equation of tangent: y โ yโ = m(x โ xโ)
Slope of normal = โ1/m
Equation of normal: y โ yโ = (โ1/m)(x โ xโ)
Increasing and Decreasing Functions
f'(x) > 0 on (a,b) โ f is strictly increasing on (a,b)
f'(x) < 0 on (a,b) โ f is strictly decreasing on (a,b)
f'(x) = 0 at a point โ possible turning point (check sign change)
f'(x) < 0 on (a,b) โ f is strictly decreasing on (a,b)
f'(x) = 0 at a point โ possible turning point (check sign change)
Maxima and Minima
First Derivative Test:
If f'(x) changes from + to โ at x = c โ local maximum
If f'(x) changes from โ to + at x = c โ local minimum
If no sign change โ neither (inflection point)
Second Derivative Test:
At critical point (f'(c) = 0):
f”(c) < 0 โ local maximum
f”(c) > 0 โ local minimum
f”(c) = 0 โ test fails (use first derivative test)
If f'(x) changes from + to โ at x = c โ local maximum
If f'(x) changes from โ to + at x = c โ local minimum
If no sign change โ neither (inflection point)
Second Derivative Test:
At critical point (f'(c) = 0):
f”(c) < 0 โ local maximum
f”(c) > 0 โ local minimum
f”(c) = 0 โ test fails (use first derivative test)
For absolute max/min on [a,b]: Find critical points in (a,b), evaluate f at critical points AND at endpoints a, b. Compare all values.
Approximations
f(x + ฮx) โ f(x) + f'(x)ยทฮx
Quick Revision Points
- Tangent slope = dy/dx; Normal slope = โdx/dy
- f'(x) > 0 โ increasing; f'(x) < 0 โ decreasing
- Critical points: where f'(x) = 0 or doesn’t exist
- 1st derivative test: check sign change of f'(x)
- 2nd derivative test: f”(c) < 0 โ max; f''(c) > 0 โ min
- Absolute extrema on closed interval: check critical points + endpoints
Chapter Navigation
Previous: Continuity and Differentiability Class 12 Notes
Next: Integrals Class 12 Notes
Related Chapters in Class 12 Maths
- Continuity and Differentiability Class 12 Notes
- Integrals Class 12 Notes
- Differential Equations Class 12 Notes
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